Question

The complex number z=x+iy which satisfies the equation $|\frac{z-5i}{z+5i}|=1$ lies on:

• A. The X-axis
• B. The line y=5
• C. A circle through the origin
• D. None of these

Answer 1

The correct option is A The X-axis

$z=x+iy$

$|\frac{z-5i}{z+5i}|=1$

$|z-5i|=|z+5i|$

$⟹|x+iy-5i|=|x+iy+5|$

$⟹\sqrt{x^2+(y-5{)}^2}=\sqrt{x^2+(y+5{)}^2}$

$⟹y^2+25-10y=y^2+25+10y$

$⟹y=0$

Hence $z$ lies on $x$ axis.